Non-volatile electric control of spin-orbit torques in an oxide two-dimensional electron gas

Spin-orbit torques (SOTs) have opened a novel way to manipulate the magnetization using in-plane current, with a great potential for the development of fast and low power information technologies. It has been recently shown that two-dimensional electron gases (2DEGs) appearing at oxide interfaces provide a highly efficient spin-to-charge current interconversion. The ability to manipulate 2DEGs using gate voltages could offer a degree of freedom lacking in the classical ferromagnetic/spin Hall effect bilayers for spin-orbitronics, in which the sign and amplitude of SOTs at a given current are fixed by the stack structure. Here, we report the non-volatile electric-field control of SOTs in an oxide-based Rashba-Edelstein 2DEG. We demonstrate that the 2DEG is controlled using a back-gate electric-field, providing two remanent and switchable states, with a large resistance contrast of 1064%. The SOTs can then be controlled electrically in a non-volatile way, both in amplitude and in sign. This achievement in a 2DEG-CoFeB/MgO heterostructures with large perpendicular magnetization further validates the compatibility of oxide 2DEGs for magnetic tunnel junction integration, paving the way to the advent of electrically reconfigurable SOT MRAMS circuits, SOT oscillators, skyrmion and domain-wall-based devices, and magnonic circuits.


S1. Determination of the AHE and PHE contributions
The AHE and PHE contributions have to be measured to quantify the SOT effective fields using Eq. (2). This is achieved by measuring the first harmonic Hall resistance, as a function of the external magnetic field applied almost in-plane (θB = 85°) and at an angle φH ≠ 0º, 90º. Fig. S1 shows RH,ω as a function of µ0Hext for an angle φH = 45° in the ungated state as an example. As AHE and PHE are respectively odd and even with respect to magnetization reversal, RAHE,ω and RPHE,ω can be separated by antisymmetrization and symmetrization of RH,ω, respectively. Fig. S1b (Fig. S1c) shows the resulting RAHE,ω (RPHE,ω) as a function of the external magnetic field (φH = 45°) for this example. The saturation value of the AHE resistance reads directly from the contrast of RAHE,ω, yielding RAHE = 25.3 Ω. The PHE saturation resistance is deduced from the linear fit of RPHE,ω as a function of sin 2 (θ) (Fig. S1d), yielding RPHE = 1.7 Ω. The equilibrium angle of magnetization (θ0) as function of the applied external magnetic field is further determined from the first harmonic Hall measurement as [1] θ 0 = arccos| R AHE,ω (µ 0 H ext ) R AHE |.

S2. Electric-control of 2DEGs with anticlockwise hysteresis in SrTiO3\\Metal\CoFeB\MgO
The electric-control of the 2DEG has been investigated in several SrTiO3\\Metal\CoFeB\MgO samples with different metallic layers (Metal) in the scope of this study. Ta, Ru, Mg(0.9 nm)\CoFeB(0.9 nm)\MgO(1.8 nm)\Ta(1 nm) stacks were deposited onto 500 µm (001)-oriented SrTiO3 substrates in order to create a 2DEG at the SrTiO3/Metal interface, following an identical sample preparation (cf. § Methods). The stacks were patterned into 1 µm wide and 10 µm wide Hall bar devices by electron-beam lithography, and sample-wide back-gates of Ti(10 nm)/Au(100 nm) were deposited by evaporation. Fig. S2 shows the sheet resistance Rs as a function of the applied gate electric-field for SrTiO3\\Mg\CoFeB\MgO and SrTiO3\\Ru\CoFeB\MgO samples, at the temperature of 10 K and after initializing the ferromagnet in the up magnetization state. Similar anticlockwise hysteresis to Figure S1 | Determination of the AHE and PHE contributions. a, First harmonic Hall resistance RH,ω as a function of the magnetic field applied almost in-plane (θH = 85°, φH = 0°, 45°, 90°). b, Antisymmetric AHE signal RAHE,ω of RH,ω(Hext,45°) yielding RAHE = 25.3 Ω. c, Symmetric PHE signal RPHE,ω of RH,ω(Hext,45°). d, RPHE,ω as a function of sin 2 (θ) showing the expected linear dependence, the slope yielding the PHE resistance RPHE = 1.7 Ω. The linear fit appears in red. All data have been measured in the ungated state at 10K with a 2D current density of j = 4 A.cm -1 . those seen in SrTiO3\\Ta\CoFeB\MgO samples are observed, with two switchable and remanent high and low resistance states. This confirms the appearance of charge trapping effect at different SrTiO3/Metal interface. The Rs contrast shows a value of 1598% (2%) for the SrTiO3\\Mg\CoFeB\MgO (SrTiO3\\Ru\CoFeB\MgO) sample. Notably, the Rs contrast of SrTiO3\\Ru\CoFeB\MgO sample is significantly lower than that of SrTiO3\\Ta,Mg\CoFeB\MgO samples, which probably indicates a poor creation of oxygen vacancies by the Ru layer at the surface of the TiO2-terminated SrTiO3 substrate. We note also the appearance of a shift of the hysteresis, which depends on the nature of the metallic layer.

S3. Separating SOT effective fields from thermal contributions
To determine the SOT effective fields, the second harmonic Hall resistance RH,2ω must be decorrelated from other contributions, corresponding to the magneto thermal contributions and to the effect of voltage probes misalignment [1] [2]. The ac current flowing into the Hall bar generates a modulated temperature gradient, which induces a second harmonic component to the Hall voltage via two types of thermoelectric voltages: the Seebeck effect and the Anomalous Nernst-Ettingshausen effect (ANE). The Seebeck effect, as well as the asymmetry of the voltage probes, induces a constant offset ROffset in RH,2ω. The ANE, on the other hand, is perpendicular to the temperature gradient and to the magnetization easy axis, and induces a second harmonic contribution proportional to the AHE in RH,2ω. Fig. S3a (Fig. S3c) shows RH,2ω as a function of µ0Hext applied at θB = 85° and φB = 0° (90°) for negative and positive gate electric-fields of ± 3.2kV.cm -1 . The ANE contribution (RANE) is observed as the difference of RH,2ω measured at zero field for positive and negative sweeps of µ0Hext, together with a constant offset (ROffset) for Eg = ± 3.2kV.cm -1 respectively. Both ROffset and RANE can be subtracted from the raw data using [1]: S3b (Fig. S3d) shows the resulting SOT anti-damping like signal (field-like signal) after subtraction of the thermal effects contributions ROffset and RANE. The data shown are taken inside the hysteresis. The arrows indicate the sweeping field direction, emphasizing the inversion of the anti-damping-like torque direction under the two different gates. Inside the hysteresis, the magnetization direction (theta angle path) is different for the positive (M↑) and negative (M↓) sweeps of magnetic field. Consequently, its susceptibility to the field-like term is different inside the hysteresis, as magnetization follows a different path, which explains the curve opening for the field-like signal. Outside the hysteresis, the curves are well closing-up, as shown in Fig. S3c. Thermal signals arising from Nernst effect are well corrected, as evidenced by the crossing close to 0 at zero-field (for in-plane thermal gradient along current direction).

S4. SiO2\\Ta\CoFeB\MgO reference sample characterization
A SiO2\\Ta\CoFeB\MgO reference sample has been prepared for comparison, with identical deposits to those made for the SrTiO3\\Ta\CoFeB\MgO samples. The stack was patterned into 1 µm wide and 10 µm long Hall cross-bars for electrical measurements. An input current of magnitude 400 A was injected along the x direction, and the longitudinal Rs and transverse Hall resistances RH were measured as a function of the external magnetic field µ0Hext. As shown in Fig. S4a, the temperature dependence of the sheet resistance of the device is characteristic of a metallic behavior, as expected from the metallic CoFeB/Ta bilayer, with a sheet resistance reaching 0.69 kΩ.sq -1 at 10K. This sheet resistance is significantly lower than that of the CoFeB/Ta bilayer in SrTiO3\\Ta\CoFeB\MgO sample, confirming that the CoFeB/Ta bilayer is partially oxidized via oxygen reduction in contact of SrTiO3. Fig. S4b shows the anomalous Hall resistance as a function of the applied out-of-plane magnetic field. A square-shaped magnetic hysteresis loop is observed, indicating that the CoFeB has a perpendicular magnetization. To quantify the spin-orbit torques, harmonic Hall voltage measurements are performed with the magnetic field µ0Hext applied almost in-plane (θH = 85°, φH = 0, 90°). Fig. S4 c and d shows the first and second harmonic Hall resistances R H,ω and R H,2ω , respectively. Following the procedure described in the main text, the anti-damping like torques and field like-torques are determined after subtracting the thermal effects contribution to R H,2ω , yielding µ0HAD/j = -1.5 mT/(A.cm -1 ) and µ0HFL/j = -0.43 mT/(A.cm -1 ). Notably, the SOT-AD effective field of the SiO2\\Ta\CoFeB\MgO reference sample is negative, of opposite sign compared to that of the SrTiO3\\Ta\CoFeB\MgO system in the low 2DEG resistivity state. This confirms that the SOT arises from the 2DEG via the Edelstein effect for SrTiO3\\Ta\CoFeB\MgO in low 2DEG resistivity state, rather than from a residual Spin Hall effect in the Tantalum layer. Note that the AD-SOT in the SiO2\\Ta\CoFeB\MgO reference  Figure S3 | Separating SOT effective fields from thermal contributions. a, Raw second harmonic Hall resistance RH,2w measured as a function of the magnetic field applied at θB = 85° and φB = 0° for negative (blue) and positive (red) gate electricfield of ± 3.2kV.cm -1 , and b, corresponding anti-damping like signal. c, Raw second harmonic Hall resistance RH,2w measured as a function of the magnetic field applied at θB = 85° and φB = 90° for negative (blue) and positive (red) gate electric-fields of ± 3.2kV.cm -1 , and d, corresponding field-like signal. All data have been measured at 10K with a 2D current density of j = 4 A.cm -1 .
sample is slightly higher than that of the SrTiO3\\Ta\CoFeB\MgO samples in high resistivity state, in agreement with the partial oxidation of the CoFeB/Ta bilayer of the latter.

S5. SOT contributions from the 2DEG and the Tantalum layer
The 2DEG and the Ta layer constitute two parallel channels of conduction, with two corresponding SOT contributions to the total SOT. In the following, we refer to HAD, 2DEG (HAD,Ta) the intrinsic SOT contribution from the 2DEG (Ta layer) alone. In a simple model, one can assume that the total SOT is the sum of the SOT contributions (in mT), weighted by the fraction of current injected in the corresponding conduction channels: HAD,signal=pHAD,2DEG +(1-p)HAD,Ta where p is the fraction of current injected in the 2DEG, displayed in Fig. 4b, and pHAD,2DEG and (1-p)HAD,Ta the 2DEG-induced and Ta-induced anti-damping like SOT, respectively. The ratio of 2DEG-induced SOT to Tainduced SOT is : As a bulk metal, the intrinsic Ta-induced SOT HAD,Ta does not depend on the gate electric-field, and is revealed directly in the high resistivity state HAD, Ta=HAD,signal(-Eg,max) when there is no current in the 2DEG (i.e p=0) and where the SOT is constant. The left panel of the Figure R3 below shows the requested ratio rAD of 2DEGinduced SOT to Ta-induced SOT at 10K, as a function of the gate electric-field. At large positive gates, the SOT is mostly due to the 2DEG, whose contribution is up to 82 times larger than that of the Ta layer. The variations of the overall SOT with the electric-field are due both to the current redistribution in the stack when varying the 2DEG resistivity, and to the variations of the conversion with the 2DEG Fermi level position. The temperature dependence of rAD in the low resistivity state can be read directly from Fig. 5b with HAD, Ta(T)=HAD,signal(-Eg,max, T), and is displayed on the right panel (for j = 2 A. cm -1 ).